Transition Matrix Models of Consumer Credit Ratings


 Lindsay Gray
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1 Transton Matrx Models of Consumer Credt Ratngs Abstract Although the corporate credt rsk lterature has many studes modellng the change n the credt rsk of corporate bonds over tme, there s far less analyss of the credt rsk for portfolos of consumer loans. However behavoural scores, whch are commonly calculated on a monthly bass by most consumer lenders are the analogues of ratngs n corporate credt rsk. Motvated by studes n corporate credt rsk, we develop a Markov chan model based on behavoural scores to establsh the credt rsk of portfolos of consumer loans. Although such models have been used by lenders to develop models for the Basel Accord, there s no publshed lterature on them. The model we suggest dffers n many respects from the corporate credt ones based on Markov chans such as the need for a second order Markov chan, the ncluson of economc varables and the age of the loan. The model s appled usng data on a credt card portfolo from a major UK bank. JEL classfcaton: C25; G21; G33 Keywords: Markov chan; Credt rsk; Logstc regresson; Credt scorng 1. Introducton Snce the md 1980s, banks lendng to consumers has exceeded that to companes ( Crouhy et al 2001). However t was only wth the subprme mortgage crss of 2007 and the subsequent credt crunch that t was realsed what an mpact such lendng had on the bankng sector and also how under researched t s compared wth corporate lendng models. In partcular the need for robust models of the credt rsk of portfolos of consumer loans has been brought nto sharp focus by the falure of the ratngs agences 1
2 to accurately assess the credt rsks of Mortgage Backed Securtes (MBS) and collateralzed debt oblgatons (CDO) whch are based on such portfolos. There are many reasons put forward for the subprme mortgage crss and the subsequent credt crunch ( Hull 2009, Demyanyk and van Hemert 2008) but clearly one reason that the former led to the latter was the lack of an easly updatable model of the credt rsk of portfolos of consumer loans. Ths lack of a sutable model of portfolo level consumer rsk was frst hghlghted durng the development of the Basel Accord, when a corporate credt rsk model was used to calculate the regulatory captal for all types of loans ( Basel Commttee on Bankng Supervson 2005) even though the basc dea of such a model that default occurs when debts exceed assets s not the reason why consumers default. Ths paper develops a model for the credt rsk of portfolos of consumer loans based on behavoural scores for the ndvdual consumers, whose loans make up that portfolo. Such a model s attractve to lenders, snce almost all lenders calculate behavoural scores for all ther borrowers on a monthly bass. The behavoural score s usually translated nto the default probablty over a fxed tme horzon ( usually one year) n the future for that borrower, but one can consder t as a surrogate for the unobservable credtworthness of the borrower. We buld a Markov chan credt rsk model based on behavoural scores for consumers whch has smlartes wth the reduced form mark to market corporate credt rsk models based on the ratng agences grades, ( Jarrow, Lando, and Turnbull 1997). Such behavoural score based Markov chan models have been developed by lenders for ther Basel modellng but no analyss has appeared n the lterature and n ths paper we dscuss what features should be ncluded n such models and compare a standard and a more sophstcated verson of the model. The methodology constructs an emprcal 2
3 forecastng model to derve a multperod dstrbuton of the default rate for long tme horzons based on mgraton matrces bult from a hstorcal database of behavoural scores. Although t s possble to calbrate score to long run probablty of default f one has data over a suffcently long outcome perod that data s not avalable n practce. The transton matrx approach allows one to undertake such calbraton usng much shorter data seres. In our case study we use the lenders behavoural scores but we can use the same methodology on generc bureau scores. The approach also helps lenders take long term lendng decsons by estmatng the rsk assocated wth the change n the qualty of portfolo of loans over tme. Snce the model ncludes economc condtons, the approach allows banks to stress test ther retal portfolos as requred by the Basel Accord and other bankng regulatons. In addton, the model provdes nsghts on portfolo proftablty, the determnaton of approprate captal reserves, and creatng estmates of portfolo value by generatng portfolo level credt loss dstrbutons. There have been some recent papers whch look at modellng the credt rsk n consumer loan portfolos. Rosch and Scheule (2004) take a varant of the one factor Credt Metrcs model, whch s the bass of the Basel Accord. They use emprcal correlatons between dfferent consumer loan types and try to buld n economc varables to explan the dfferences durng dfferent parts of the busness cycle. Perl and Nayda (2004) also take the corporate credt rsk structural models and seek to apply t to consumer lendng assumng that a consumer defaults f hs assets are lower than a specfed threshold. However consumer defaults are usually more about cash flow problems, fnancal naveté or fraud and so such a model msses some of the aspects of consumer defaults. 3
4 Musto and Souleles (2005) use equty prcng as an analogy for changes n the value of consumer loan portfolos. They use behavoural scores but take the monthly dfferences n behavoural scores as the return on assets when applyng ther equty model. Andrade and Thomas ( 2007) descrbe a structural model for the credt rsk of consumer loans where the behavoural score s a surrogate for the credtworthness of the borrower. A default occurs f the value of ths reputaton for credtworthness, n terms of access to further credt drops below the cost of servcng the debt. Usng a case study based on Brazlan credt bureau they found that a random walk was the best model for the dosyncratc part of credtworthness. Malk and Thomas (2010) developed a hazard model of tme to default for consumer loans where the rsk factors were the behavoural score, the age of the loan and economc varables, and used t to develop a credt rsk model for portfolos of consumer loans. Bellott and Crook ( 2009) also used proportonal hazards to develop a default rsk model for consumer loans. They nvestgated whch economc varables, mght be the most approprate though they dd not use behavoural scores n ther model. Thomas (2009b) revewed the consumer credt rsk models and ponted out the analoges wth some of the establshed corporate credt rsk models. Snce the semnal paper by Jarrow, Lando and Turnbull ( Jarrow et al 1997), the Markov chan approach has proved popular n modellng the dynamcs of the credt rsk n corporate portfolos. The dea s to descrbe the dynamcs of the rsk n terms of the transton probabltes between the dfferent grades the ratng agences award to the frm s bonds. There are papers whch look at how economc condtons as well as the ndustry sector of the frm affects the transtons matrces, ( Nckell et al 2001) whle others generalse the orgnal Jarrow, Lando Turnbull dea by usng Affne Markov chans 4
5 (Hurd and Kuznetsov 2006) or contnuous tme processes ( Lando and Skodeberg 2002). However none of these suggest ncreasng the order of the Markov chan or consderng the age of the loan whch are two of the features whch we ntroduce n order to model consumer credt rsk usng Markov chans. Ths s surprsng because there s work on downgradng by ratng agences, whch suggests there s a momentum effect n whch when a company has been downgraded t s more lkely to be further downgraded than to be subsequently upgraded ( Banga et al 2002). Markov chan models have been used n the consumer lendng context before, but none of the publshed papers use the behavoural score as the state space nor s the objectve of the models to estmate the credt rsk at the portfolo level. The frst applcaton was by Cyert (1962) who developed a Markov chan model of customer s repayment behavour. Subsequently more complex models have been developed by Ho (2001), Thomas et al (2001) and Trench et al (2003). Schnederjans and Lock (1994) used Markov chan models to model the marketng aspects of customer relatonshp management n the bankng envronment. Behavoural score based Markov chan models are sometmes used n the ndustry, see Scallan (1998) but manly as ways of assessng provsonng estmates and they do not nclude the economc drvers and months on books effects presented n ths paper. Moreover the ntroducton of economc factors nto the model allows one to deal wth the correlatons between defaults on ndvdual loans n a portfolo snce they are affected by common economcs. One can get the mean default rate n a portfolo from the long run dstrbutons whle a Monte Carlo smulaton usng the transtons of ndvdual loans would gve the dstrbuton of the default rate. 5
6 In secton two, we revew the propertes of behavoural scores and Markov chans, whle n secton three we descrbe the Markov chan behavoural score based consumer credt rsk model developed. Ths s parametersed by usng cumulatve logstc regresson to estmate the transton probabltes of the Markov chan. The motvaton behnd the model and the accuracy of the model s forecasts are gven by means of a case study and secton four descrbes the detals of the data used n the case study. Sectons fve, sx and seven gve the reasons why one ncludes n the model hgher order transton matrces (secton fve); economc varables to explan the non statonarty of the chan (secton sx) and the age of the loan (secton seven). Secton eght descrbes the full model used, whle secton nne reports the results of out of sample and out of tme and out of sample forecasts usng the model. The fnal secton draws some conclusons ncludng how the model could be used. It also dentfes one ssue whch economc varables drve consumer credt rsk where further nvestgaton would beneft all models of consumer credt rsk. 2. Behavour Score Dynamcs and Markov Chan models Consumer lenders use behavoural scores updated every month to assess the credt rsk of ndvdual borrowers. The score s consdered to be a suffcent statstc of the probablty a borrower wll be Bad and so default wthn a certan tme horzon (normally taken to be the next twelve months). Borrowers who are not Bad are classfed as Good. So at tme t, a typcal borrower wth characterstcs x(t) ( whch may descrbe recent repayment and usage performance, the current nformaton avalable at a credt bureau on the borrower, and socodemographc detals) has a score s(x(t),t) so 6
7 p( B x( t), t) = p( B s( x( t), t)) (1) Some lenders obtan a Probablty of Default (PD), requred under the Basel Accord by takng a combnaton of behavoural and applcaton scores. New borrowers are scored usng only the applcaton score to estmate PD: once there s suffcent hstory for a behavoural score to be calculated, then a weghted combnaton of the two scores s used to calculate PD; eventually the loan s suffcently mature that only the behavoural score s used to calculate PD. The models descrbed hereafter can also be appled to such a combned scorng system. Most scores are log odds score (Thomas 2009a) so the drect relatonshp between the score and the probablty of beng Bad s gven by P( G s( x( t), t) 1 s( x( t), t) = log P( B s( x( t), t)) = P( B s( x( t), t) 1+ e s( x( t ), t ) (2) though n realty ths may not hold exactly. Applyng Bayes theorem to (2) gves the expanson where f p G (t) s the proporton of the populaton who are Good at tme t (p B (t) s the proporton who are Bad) one has P( G s( x( t), t) pg ( t) P( s( x( t), t) G, t) s( x( t), t) = log = log + log = spop ( t) + woe t( s( x( t), t)) (3) P( B s( x( t), t) pb( t) P( s( x( t), t) B, t) The frst term s the log of the populaton odds at tme t and the second term s the weght of evdence for that score, (Thomas 2009a). Ths decomposton may not hold exactly n practce and s lkely to change as a scorecard ages However t shows that the term s pop (t), common to the scores of all borrowers, can be thought to play the role of a systemc factor whch affects the default rsk of all the borrowers n a portfolo. Normally though the tme dependence of a behavoural score s gnored by lenders. Lenders are usually only nterested n rankng borrowers n terms of rsk and they beleve that the second term 7
8 ( the weght of evdence ) n (3), whch s the only one that affects the rankng, s more stable over tme than s pop (t) partcularly over horzons of two or three years. However the tme dependence s mportant because t descrbes the dynamcs of the credt rsk of the borrower. Gven the strong analoges between behavoural scores n consumer credt and the credt ratngs used for corporate credt rsk, one obvous way of descrbng the dynamcs of behavoural scores s to use a Markov chan approach smlar to the reduced form mark to market models of corporate credt rsk (Jarrow et al 1997). To use a Markov chan approach to behavoural scores, we dvde the score range nto a number of ntervals each of whch represents a state of the Markov chan, and hereafter when we menton behavoural scores we are thnkng of ths Markov chan verson of the score, where states are ntervals of the orgnal score range. Markov chans have proved ubqutous models of stochastc processes because ther smplcty beles ther power to model a varety of stuatons. Formally, we defne a dscrete tme {t 0,t 1,...,t n,...: n N} and a fnte state space S = {1,2,...,s} frst order Markov chan as a stochastc process {X(t n )} n N wth the property that for any s 0, s 1,,s n1,, j S P[ X ( tn+ 1 ) = j X ( t0 ) = s 0,X ( t1 ) = s 1,...,X ( tn 1 ) = s n 1,X ( tn ) = ] = P[ X ( t ) = j X ( t ) = ] = p ( t,t ) ( ) n+ 1 n j n n+ 1 4 where p j (t n,t n+1 ) denotes the transton probablty of gong from state at tme t n to state j at tme t n+1. The s s matrx of elements p j (.,.), denoted P(t n,t n+1 ), s called the frst order transton probablty matrx assocated wth the stochastc process {X(t n )} n N. If π ( t ) = ( π ( t ),..., π ( t )) descrbes the probablty dstrbuton of the states of the n 1 n s n process at tme t n, the Markov property mples that the dstrbuton at tme t n+1 can be 8
9 obtaned from that at tme t u by π ( t ) = π ( t ) P ( t,t ). Ths extends to a mstage n+ 1 n n n+ 1 transton matrx so that the dstrbuton at tme t n+m for m 2s gven by π ( t ) = π ( t ) P( t,t )... P ( t,t ) n+ m n n n+ 1 n+ m 1 n+ m The Markov chan s called tme homogeneous or statonary provded pj ( t n,tn+ 1 ) = pj n N. ( 5) Assume the process {X(t n )} n N s a nonstatonary Markov chan, whch s the case wth the data we examne later. If one has a sample of hstores of prevous customers, let n (t n ), S, be the number who are n state at tme t n, whereas let n j (t n,t n+1 ) be the number who move from state at tme t n to state j at tme t n+1. The maxmum lkelhood estmator of p ( t,t + ) s then j n n 1 nj ( t n,tn+ 1 ) ˆp j ( t n,t n+ ) =. n ( t ) 1 6 n If one assumed that the Markov chan was statonary, then gven the data for T+1 tme perods n= 0, 1, 2,, T, the Transton probablty estmates become ( ) ˆp j = T 1 n= 0 n ( t,t ) j n n+ 1 T 1 n= 0 n ( t ) n ( 5) Note that the Markov property means that prevous transtons do not affect the current probabltes of transton and so n these calculatons we do not need to be concerned that transtons comng from the same customer are dependent. All transtons are essentally ndependent even those from the same customer. One can weaken the Markov property so that the nformaton requred to estmate the future of the chan s the current state and the prevous state of the process. Ths s called a second order Markov chan whch s 9
10 equvalent to the process beng a frst order Markov chan but wth state space S S. The concept can be generalzed to defnng k th order Markov chans for any k, though of course, the state space and the sze of the transton probablty matrces goes up exponentally as k ncreases. 3. Behavoural score based Markov Chan model of Consumer Credt Rsk The behavoural score B t of a borrower s an observable varable gven by a scorecard. It s related to the underlyng unobservable credt worthness, U t of the borrower, whch also depends on the length of tme the loan has been runnng and the current economc stuaton. Our model s constructed by assumng that the borrower s behavoural score s n one of a fnte number of states, namely {s 0 =D, s 1, s n, C} where s >0 descrbes an nterval n the behavoural score range; s 0 =D means the borrower has defaulted and C s the state when the borrower closed hs loan or credt card account havng repad everythng ( an absorbng state). The Markov property means that the dynamcs from tme t onwards of the behavoural score s condtonal on the realzaton of the score state at tme t1, B t1 or at least that ts movement between the score range ntervals depends only on whch current nterval t s n. Gven the behavoural score s n state s, =1,.n, at tme t1, we wrte the latent varable U t at tme t as defned so that the relatonshp between B t and U t s that U t. For the actve accounts, U t s B = s µ U µ, j = 0,1,.. n wth µ =, µ = (6) t j j t j+ 1 0 n+ 1 where µ j are the values n the unobservable credt worthness whch correspond to the end ponts of the behavoural score ntervals s. Moreover one chooses µ 1 so that f the 10
11 consumer defaults one must have U t µ 1. The dynamcs of the underlyng varable U t s assumed to be related to the explanatory varable vector x t1 by a lnear regresson of the ' form Ut = β xt 1 + ε t, where β s a column vector of regresson coeffcents and ε t are random error terms. If the ε t are standard logstc dstrbutons, then ths s a cumulatve logstc regresson model and the transton probabltes of B t are gven by ' ( t = t 1 ) = ( µ 1 + β xt 1 ) Prob B D B =s logt, ' ' ( = ) = ( µ + β x ) ( µ + β x ) ( ) Prob B s B =s logt logt, 7 t 1 t 1 2 t 1 11 t 1 Prob( B = s B =s ) = 1 logt µ + β t n t 1 n ' ( t 1 ) x. Estmatng cumulatve logstc model usng usual maxmum lkelhood means that condtonal on the realzaton the tme dependent covarate vector x t1, transtons to varous states for dfferent borrowers n the next tme perod are ndependent both crosssectonally and through tme. So the dynamcs of the behavoural scores s drven by the explanatory varable x t1. In the model presented we assume three types of drvers economc varables, the age of the loan and the prevous behavour of the score. We justfy these choces n sectons 5 to 7 by lookng at ther effect on the smple frst order Markov chan model. Note that states C and D are absorbng states and so there are no transtons from them and we wll dscuss the modelng of movements to the closed state, C, n secton 8 Ths has parallels wth some of the corporate credt rsk models. In Credt Metrcs for example (Gordy 2000) the transton n corporate ratngs are gven by changes n the underlyng asset varables n a smlar fashon but wth qute dfferent drvers. 11
12 Snce behavour scores are only calculated monthly, calendar tme t needs to be dscrete and then the credtworthness at tme t of a borrower, whose credt worthness at tme t1 was n state,s gven by the latent varable U t, whch satsfes the relatonshp K t = k t k b EcoVar t1 t 1 + εt k = 2 U a State. c MoB ( 8) where State tk s a vector of ndcator varables denotng borrower s state at tme tk, EcoVar t1 s a vector of economc varables at tme t1, MoB t1 s a vector of ndcator varables denotng the length of tme the loan has been on the books n months ( Months on Books) at tme t1. One could smooth ths latter effect by usng a contnuous varable of the age of the loan but we descrbe the effect usng more predctve bnary varables for dfferent age bands. a, b, and c are coeffcents n the expresson and ε t s a random varable representng a logt error term. Snce U t depends on, the underlyng credtworthness at tme t depends on the state at t1 and so the behavoural score at tme t wll also depend on the state, and hence the behavoural score, at tme t1. If ak 0 then the credt worthness at tme t also depends on the state at tme tk and so the Markov chan model of the correspondng behavoural scores B t wll be of order k. The transtons also depend on economc varables and on the length of tme the loan has been repad. Snce the coeffcents depend on then the mpact of these other factors wll vary from state to state. If the score band ntervals were of equal length and the decomposton n (3) really held then one would expect a = 0, c = 0, b = b and so ths model allows for more complex dynamcs n the behavoural scores. k The Months on books term does not occur n any corporate credt models, but s of real mportance n consumer lendng ( Breeden 2007, Stepanova and Thomas 2002). Smlarly 12
13 t s rare to have hgher order Markov chans models n corporate credt, though the state space s sometmes extended to nclude whether there have been recent upgrades or downgrades n the ratngs. Thus although corporate credt models may have more complex factors affectng ther dynamcs such as ndustry type, geographcal area and senorty of the debt, they are not so much affected by recent changes of state or the age of the loan whch are mportant n consumer credt rsk models. 4. Data Descrpton The dataset used for the case study n ths paper contans records of credt card customers of a major UK bank who were on the books as of January 2001 together wth all those who joned between January 2001 and December The data set conssts of customers' monthly behavoural scores along wth the nformaton on ther tme snce account opened, tme to default or tme when the account was closed wthn the above duraton. We randomly selected approxmately 50,000 borrowers for a tranng data set whch contaned ther hstory over the perod Jan 2001 Dec We tested our Markov models usng customer s performance durng 2005 from a subsample of the 50,000 and also from a holdout sample of approxmately 15,000 customers. Anyone, who became 90 days delnquent (even f they subsequently were cured), was charged off or declared bankrupt, s consdered as havng defaulted. The bank reported that there were no major changes n credt lmt settng or mnmum repayment levels durng the perod under consderaton, nor were there any changes n the scorecard or ntentonal attempts to change the mx of the portfolo of borrowers 13
14 through portfolo acquston or marketng campagns. To analyse the changes n the dstrbuton of behavoural score we frst coarse classfy behavoural score nto varous segments. Intally, we segment the behavoural score nto decles of the dstrbuton of the score among all the borrowers n the sample over all the months n the sample. We use the chsquare statstc to decde whether to combne adjacent decles f ther transton probabltes are suffcently smlar. Ths technque of coarse classfyng s standard n scorecard buldng (Thomas 2009a) to deal wth contnuous varables where the relatonshp wth default s non lnear. In ths case t led to a reducton to fve scorebands, namely s 1 ={ }, s 2 ={ }, s 3 ={ }, s 4 ={ } and s 5 ={726 and above}. As well as these fve states there are two more specal states correspondng to Default and Account Closed. If there are too many states n the chan the parameter estmates lose robustness, whle f there are too few one loses structure and one does not have enough segments to valdate the model accordng to the Basel Accord requrements. Behavoural scores are generated or updated every month for each ndvdual so t would be possble to estmate a 1month tme step transton matrx. Snce transtons between some states wll have very few 1 month transtons, such a model may lead to less than robust estmates of the parameters. Hence we use 3month tme steps. Longer tme steps, say sx or twelve months, make t harder to nclude the mpact of the changes n economcs and the months on books effect. In the followng sectons we shall justfy the use of hgher order Markov chans and provde an analyss of the effects of tme varyng macroeconomc and months on books covarates on behavoural score transtons. 14
15 5. Order of the Transton Matrx We frst estmate the average transton matrx, assumng the Markov chan s statonary and frst order usng the whole duraton of the sample from January 2001 to December Table 1 shows the 3month tme step transton matrx for that sample, where the fgures n brackets are the standard samplng errors. As one mght expect, once a borrower s n the least rsky state ( s 5 ) there s a hgh probablty, 88%, they wll stay there n the next quarter. More surprsngly the state wth the next hghest probablty of the borrower stayng there s s 1, the rskest behavoural score state, whle borrowers n the other states move around more. The probabltes of defaultng n the next quarter are monotone wth, as one would expect, beng the most rsky state wth a default probablty of 6.7% and 726hgh the least rsky state wth a default probablty of 0.2%. Note that there s the obvous stochastc domnance ( p p + 1 ) for all the actve j k j j j k states, whch shows that the behavoural score correctly reflects future score changes as well as future defaults. Table 1: Frst Order Average Transton Matrx Intal State Transton State hgh Closed Default (0.2) (0.2) (0.1) (0.1) (0.1) (0.1) (0.1) (0.1) (0.2) (0.2) (0.1) (0.1) (0.1) (0.0) (0.1) (0.1) (0.2) (0.1) (0.1) (0.1) (0.0) (0.1) (0.1) (0.1) (0.2) (0.2) (0.1) (0.0) 726hgh (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) Ths frst order Markov chan model assumes that the current state has all the nformaton needed to estmate the probablty of the transtons next quarter and so these are unaffected by the borrower s prevous states. If ths s not true, one should use a second or hgher order Markov chan model. Ths mght seem surprsng n that a behavoural 15
16 score s consdered to be a suffcent statstc of the credt rsk. However ths s a very specfc credt rsk the chance of default n the next 12 months whereas the Markov chan descrbes the dynamcs of the credt rsk estmates over a dfferent 12 month nterval each perod. Thus t s qute possble the score does not nclude all the nformaton needed to estmate how ths rsk s lkely to change. Table 2 dsplays the estmates of the transton matrx for such a second order chan, obtaned n a smlar way as Table 1. Analysng Table 2 shows that there are substantal changes n the transton probabltes based on the prevous state of the borrower. Consder for example f the current state s the rsky one s 1 = {13680}. If borrowers were also n the rsky state last quarter then the chance of stayng on t or defaultng n the next quarter s 58% +7%=65%.; f they were n the least rsky state n the last quarter { 726+} but are now n s 1, the chance of beng n s 1 or default next quarter s 22.8%+7.7%=30.5%. Table 2: Second Order Average Transton Matrx (Prevous State, Current State) Termnal State hgh Closed Default (13680,13680) ( ,13680) ( ,13680) ( ,13680) (726hgh,13680) (13680, ) ( , ) ( , ) ( , ) (726hgh, ) (13680, ) ( , ) ( , ) ( , ) (726hgh, ) (13680, ) ( , ) ( , ) ( , ) (726hgh, ) (13680,726hgh) ( ,726hgh) ( ,726hgh) ( ,726hgh) (726hgh,726hgh)
17 So there s a propensty to reverse drecton and return n the drecton one came. Ths effect s seen n all the fve behavoural score nterval states n the model. These results do not support the momentum dea that borrowers whose score has dropped are more lkely to drop further (see Banga et al 2002 for examples n corporate credt), but suggests there may be some event of very short duraton whch appears and then s reversed n the next quarter, such as beng put n arrears due to some msunderstandng. Ths effect seen n all fve states could be due to usng score bands rather than the scores themselves and so the prevous score band mght suggest where n the nterval the score s. However the same result was seen when a fner classfcaton,.e. more states wth smaller ntervals, was used. One could nvestgate whether hgher order models are even more approprate but for thrd and hgher order Markov chans data sparsty and robustness of predctons become problems and so we use a second order chan to model the dynamcs of the behavoural scores. 6. Macro Economc Varables Tradtonally behavoural score models are bult on customers performance wth the bank over the prevous twelve months usng characterstcs lke average account balance, number of tmes n arrears and current credt bureau nformaton. So the behavoural score can be consdered as capturng the borrower s specfc rsk. However, n corporate credt rsk models (Das et al, 2007), t was shown that though borrower specfc rsk s a major factor, durng economc slowdowns systemc rsk factors emerge and have had a substantal effect on the default rsk n a portfolo of loans. The decomposton of the behavoural score n (3) suggests ths s also the case n consumer lendng, snce the 17
18 populaton log odds s pop (t) must be affected by such systemc changes n the economc envronment. The queston s whch economc varables affect the default rsk of consumers. We nvestgate fve varables whch have been suggested as mportant n consumer fnance ( Tang et al 2007, Lu and Xu 2003), together wth one varable that reflects market condtons n consumer lendng. The varables consdered are: (a) Percentage Change n Consumer Prce Index over 12 Months: reflects the nflaton felt by customers and hgh levels may cause rse n customer default rate. (b) Monthly average Sterlng Interbank lendng rate: hgher values correspond to general tghtness n the economy as well as ncreases n debt servce payments. (c) Annual Return on FTSE 100: gves the yeld from stock market and reflects the buoyancy of ndustry. (d) Percentage change n GDP compared wth equvalent Quarter n Prevous Year: (e) UK Unemployment Rate. (f) Percentage Change n Net Lendng over 12 Months: ths gves an ndcaton of the funds beng made avalable for consumer lendng. There s a general percepton (Fglewsk et al, 2007) that change n economc condtons do not have an nstantaneous effect on default rate. To allow for ths, we use lagged values of the macroeconomc covarates n the form of weghted average over a sx months perod wth an exponentally declnng weght of Ths choce s motvated by the recent study made by (Fglewsk et al, 2007). Snce macro economc varables represent the general health of the economy they are expected to show some degree of correlaton. Table 3 below shows the parwse correlaton matrx for the above sx 18
19 macroeconomc varables wth no lags consdered. The entres n bold are the correlatons consdered statstcally sgnfcant at the 5% level. Thus at a 5% sgnfcance level nterest rate s negatvely correlated wth percentage change n CPI and postvely correlated wth percentage change n GDP and return on the FTSE 100. Smlarly, percentage change n Net Lendng s negatvely correlated wth Unemployment rate and postvely correlated wth percentage change n GDP and return on the FTSE 100 at 5% sgnfcance level. The presence of non zero correlaton between varables does not nvaldate the model, but the degree of assocaton between the explanatory varables can affect parameter estmaton. Moreover the varables used are chosen n so as to avod long run trends and the fact that three of the varables are percentage changes s akn to already takng dfferences to avod non statonarty Table 3: Correlaton matrx of macroeconomc factors Interest % change n % change n % change n unemployment Return on Rate CPI GDP net lendng rate FTSE 100 Interest Rate % change n CPI % change n GDP ` % change n net lendng unemployment rate Return on FTSE Fgure 1 shows the varaton of the observed log(default Odds) over 3 month wndows compared wth the lagged macroeconomc factor values used n the analyss for the 19
20 sample duraton of January 2001 to December The macroeconomc factors values are represented by the prmary yaxs and the log(default Odds) by the secondary yaxs. Fgure 1:3Month Observed log(odds Default) and Macroeconomc varables 25.0% % % % 15.0% 4.6 UR 12.5% 4.8 Int Rate CPI 10.0% 7.5% 5.0% FTSE 100 Net Lendng GDP logodds 2.5% 0.0% % Jan01 Jun01 Nov01 Apr02 Sep02 Feb03 Jul03 Dec03 May % 6 We plot the lagged economc values for each month though of course we only use the values every quarter n the Markov chan model snce ts tme perod s quarterly. In the bengn envronment of there are no large swngs n any varable and the log of the default odds  s pop (t) s qute stable. To convnce ourselves that changes n economc condtons do affect the transtons matrx, we look at transton matrces based on data from two dfferent tme perods, whch have slghtly dfferent economc condtons. In order not to complcate matters we show the dfferences that occur even n the frst order Markov chan. In Table 4, we estmate the frst order transton probablty matrces for two dfferent twelve months calendar tme perods between Jan 2001 to December 2004 to judge the effect of calendar tme on transton probabltes. The frst matrx s based on sample of customers who were on books durng JanDec 2001 and uses ther transtons each quarter durng that 20
21 perod and the second s based on those n the portfolo durng Sept03 Oct04 and ther performance durng that perod. Both transton matrces show consderable smlartes wth the whole sample average transton matrx n Table 1, wth the probablty of movng nto default decreasng as the behavoural score ncreases and the stochastc domnance effect stll holdng. However there are some sgnfcant dfferences between the transton probabltes of the two matrces n Table 4. For example, borrowers who were n current state of s 1 ={13680} durng JanDec 2001 have a lower probablty of defaultng n the next quarter 5.5%  than those who were n the same state Table 4: Comparson of transton matrces at dfferent calendar tmes Intal state Termnal State Closed Default number n state JanDec Oct 03Sept
22 durng Sept03 Oct04 where the value s 8.22%. We test the dfference between the correspondng transton probabltes n the two matrces n Table 4 usng the twoproporton ztest wth unequal varances. The entres n bold n Table 4 dentfy those transton probabltes where the dfferences between the correspondng terms n the two matrces are sgnfcant at the 5% level. Note that there are 35 transton probabltes beng compares and so one mght expect 2 sgnfcant comparons at the 5% level f there were really no dfference. There are 20 sgnfcant dfferences whch suggest ths calendar effect s real. 7. Months on Books Effects As s well known n consumer credt modelng (Breeden 2007, Stepanova and Thomas 2002), the age of the loan (the number of months snce the account was opened) s an mportant factor n default rsk. To nvestgate ths we splt age nto seven segments namely, 06 months, 712 months, months, months, months, months, more than 48 months.. The effect of age on behavoural score transton probabltes can be seen n Table 5, whch shows the frst order probablty transton matrces for borrowers who were on books between one to twelve months ( upper table) and more than 48 months ( lower table). Agan the overall structure s smlar to Table 1, but there are sgnfcant dfferences between the transton probabltes of the two matrces. Borrowers who are new on the books are more at rsk of defaultng or of ther behavoural score droppng than those who were wth the bank for more than four years. The bold entres agan represent transtons where the dfferences between the new and mature accounts are sgnfcantly dfferent at the 5% level.4n the Agan the fnal block of 22
23 Table 5 gves the z statstc and the bold values ndcate where n the tables the dfferences n transtons are statstcally sgnfcant at the 5% level. Ths occurs n 27 out of the 35 transtons calculated Table 5: Comparson of transton matrces for loans of dfferent ages Intal state Termnal State Closed Default number n state 112 months ( new oblgors) hgh( mature oblgors) Modelng Transton Probabltes Behavoural score segments have a natural orderng structure wth low behavoural score assocated wth hgh default rsk and vce versa. Ths s the structure that s exploted 23
24 when usng cumulatve (ordered) logstc regresson to model borrowers' transtons probabltes as suggested n secton 3. (McElvey and Zavona, 1975). The cumulatve logstc regresson model s approprate for modellng the movement between the behavoural scorebands and the defaulted state. If we wshed also to model whether the borrowers close ther accounts one would need to use a two stage model. In the frst stage, one would use logstc regresson to estmate the probablty of the borrower closng the account n the next quarter gven hs current state, P(Close beh.score band). The second stage would be the model presented here of the movement between the dfferent scorebands ncludng default condtonal on the borrower not closng the account. To arrve at the fnal transton probabltes one would need to multply the probabltes for each transton obtaned n ths second stage by the chance the account s not closed obtaned from the frst stage, (1P(Close beh.score band)). Ths approach assumes the resduals of the estmatons n the two stages are ndependent. So we now ft the cumulatve logstc model to estmate the transton probabltes of a borrower s movement n behavoural score from beng n state at tme t1 B = t 1 s to where the borrower wll be at tme t, B t. These transtons depend on the current state B t1 = s ( snce they are ndexed by ), the prevous state of the borrower, Bt 2, the lagged economc varables and the age of the loan ( Months on Books or MoB). So one uses the model gven by (6) and (8) but restrcted to the second order case, namely B = s µ U µ, j = 0,1,.. n wth µ =, µ = t j j t j+ 1 0 n+ 1 U = a State b EcoVar c MoB + ε t t 2 t 1 t 1 t (9) In order to choose whch economc varables to nclude, we recall that Table 3 descrbed the correlaton between the varables. To reduce the effect of such correlatons (so that 24
25 the coeffcents of the economc varables are understandable), we consdered varous subsets of the macro economc varables as predctors n a cumulatve logstc model, where there was lttle correlaton between the varables. In Table 6 we present parameter estmates for the cumulatve logstc models for each behavoural score segment wth only two macroeconomc varables, namely nterest rate and net lendng, along wth months on books and the prevous state. Ths means we allow the drvers of the dynamcs economc varables and current duraton of loan to have dfferent effects on the transtons from dfferent states. The model wth these two varables nterest rate and net lendng provded a better ft n terms of the lkelhood rato of the model than other combnatons of macroeconomc varables the next best ft was unemployment and nterest rates. We employ stepwse selecton keepng only varables wth a 5% sgnfcance level for the correspondng regresson coeffcent to be nonzero. The lkelhood ratos and the assocated pvalues show that for each current behavoural score segment, transtons to other states n the next tme perod are sgnfcantly nfluenced by current macroeconomc factors, current months on books and nformaton on prevous state, represented by a Secstate varable n Table 6. Ths model fts the data better than the frst order average transton matrx. A postve sgn of the coeffcent n the model s assocated wth a decrease n credtworthness and vce versa. So the credtworthness of borrowers decreases n the next tme perod wth an ncrease n nterest rates n all current behavoural score segments. Borrowers who are between 7 and 18 months on the books have hgher default and downgradng rsks than the others. Ths confrms the market presumpton that new borrowers have hgher default rsk than older borrowers n any gve tme perod, once 25
26 they have had suffcent tme (.e at least 3 months) to default. The coeffcents of the Secstate varable, wth one excepton, decrease monotoncally n value from the s 1 ={13680} category to the s 5 ={726hgh} state. Those wth lower behavoural score last quarter are more lkely to have lower behavoural score next quarter than those wth the same behavoural score currently but who came from hgher behavoural score bands. So the dea of credt rsk contnung n the same drecton s not supported. Table 6: Parameters for second order Markov chan wth age and economc varables Parameter Estmates Intal Behavoural Score Std Error Std Error Std Error Std Error 726hgh Std Error Interest Rate (0.0161) (0.0143) (0.0123) (0.0134) ( ) Net Lendng ( ) Months on Books (0.0351) (0.0347) (0.0368) (0.0448) (0.0315) (0.0241) (0.0225) (0.0226) (0.0261) (0.018) (0.0262) (0.0236) (0.0228) (0.0268) (0.0193) (0.0275) (0.0251) (0.0241) (0.0284) (0.0206) (0.0244) (0.0223) (0.0198) (0.0229) (0.0162) (0.0262) (0.0243) (0.0206) (0.0241) (0.0166) 49hgh SecState (0.0165) (0.0168) (0.0222) (0.0337) (0.0399) (0.0201) (0.0139) (0.0146) (0.0224) (0.024) (0.0249) (0.0168) (0.0119) (0.0164) (0.0178) (0.0345) (0.0239) (0.016) (0.0152) (0.0162) 726hgh Intercept/Barrer Default (0.0756) (0.0826) (0.1285) (0.0967) (0.0506) (0.0734) (0.0657) (0.1175) (0.0648) (0.0436) (0.0736) (0.0649) (0.1168) (0.062) (0.0421) (0.0746) (0.065) (0.1165) (0.0611) (0.0415) (0.0764) (0.0656) (0.0609) (0.0413) Lkelhhod Rato Pvalue < < < < < Forecastng MultPerod Transton Probabltes The model wth the parameters gven n Table 6 was tested by forecastng the future dstrbutons of the scorebands n the portfolo, ncludng those who have defaulted. The 26
27 forecast uses the Markov assumpton and so multples the probablty transton matrx by tself the approprate number of tmes to get the forecasts. In the frst case we consder all nondefaulted borrowers n December 2004 and used the model to predct ther dstrbuton over the varous behavoral score bands and the default state at the end of each quarter of 2005, where closures were dealt wth as descrbed n secton 8. Not to add extra uncertanty to the forecast, the 2005 values of the two economc varables were used. The results are shown n Table 7. The ntal dstrbuton column gves the dstrbuton of borrowers nto each behavoural score segment n the test sample n December The observed column gves the observed dstrbuton of borrowers at the end of each quarter n The other two columns gves the expected number of borrowers n each segment at the end of each quarter of 2004 as predcted by the second order average transton matrx n Table 2 and those predcted by the model n Table 6. Table 7: Dstrbuton at the end of each tme perod on out of tme sample test sample (2005) Behavoural Score 1Perod 2Perod 3Perod 4Perod Segments Intal Dstrbuton Average Matrx Model Predcted Observed Average Matrx Model Predcted Observed Average Matrx Model Predcted Observed Average Matrx Model Predcted Observed hgh Default The second order Markov chan model wth economc varables gave predctons, partcularly for defaults, whch were very close to the actual values for the frst and second quarters, but begn to overestmate the rsks thereafter. So by the fourth quarter the average second order Markov chan model whch just takes the average of the transton probabltes s superor. The analyss was repeated on an out of tme and out of sample portfolo. Agan the dstrbuton of the portfolo at the start of the perod (Aprl 2005) was gven and estmates 27
28 for the next three quarters obtaned usng the model n Table 6. The results n Table 8 show that the second order model wth economc and months on books effect (Table 6) s better at predctng the actual number of defaults than the second order model wthout these effects (Table 3) even though both approaches slghtly under predct. The model wth the extra drvers s better at predctng the numbers n the default and hgh rsk states, whle the second order one that just averages over all transtons s better at predctng the numbers n the low rsk categores. In ths data set t appears the second order effect s the most mportant followed by the Months on books effect. However ths could be due to the relatve economc stablty throughout both the perod represented by both the development sample and the out of tme test sample. Table 8 Dstrbuton at the end of each tme perod on out of tme out of sample test sample (2005) Behavoural Score 1Perod 2Perod 3Perod Segments Intal Dstrbuton Average Matrx Model Predcted Observed Average Matrx Model Predcted Observed Average Matrx Model Predcted Observed hgh Default Conclusons The paper has developed a plot scheme on how one could use a Markov chan approach based on behavoural scores to estmate the credt rsk of portfolos of consumer loans. Ths s an attractve approach snce behavoural scores are calculated monthly by almost all lenders n consumer fnance, both for nternal decson purposes and for Basel Accord requrements. The paper emphasses that behavoural scores are dynamc and snce they do have a systemc factor the populaton odds part of the score the dynamcs depends on changes n economc condtons. The paper also suggests one needs to 28
29 consder carefully the approprate order of the Markov chan. Table 2 shows the mpact of the prevous state as well as the current state on the subsequent transton and strongly suggests the need for a second order Markov chan. Unlke corporate credt rsk, one also needs to nclude the age of the loan nto the modellng as ths affects the credt rsk. The out of sample comparson of second order models wth and wthout economc factors and age n the model are nconclusve about whch model s better but ths s a tme when the economc condtons were very stable. In more volatle condtons or f one wants to use the model for stress testng then t wll be essental to nclude the economc effects nto the modellng. Such models are relatvely easy for banks to develop snce they have all the nformaton readly avalable. The model would be useful for a number of purposes debt provsonng estmaton, stress testng n the Basel context as well as nvestgatng the relatonshp between Pont n Tme Behavour Scores and through the cycle probabltes of default by runnng the model through an economc cycle. The model could also be used by ratngs agences to update ther rsk estmates of the securtzed products based on consumer loan portfolos. Ths would requre then to obtan regular updates of the behavoural scores of the underlyng loans rather than the present approach of just makng one ntal ratng based on an applcaton or bureau score. Ths s extra work but mght avod the falures of the ratng of the mortgage backed securtes (MBS) seen n 2007 and 2008 and would certanly gve early warnng of the ncreasng credt rsk n such securtes. 29
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